The combinatorics of the colliding bullets problem

The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let $(V_i)_{1\le i\le n}$ be an i.i.d.\ family of random variables with uniform distribution on $[0,1]$; shoot $n$ bullets one after another at times $1,2,\dots, n$, where the $i$th bullet has speed $V_i$. When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalisation of this model. While the distribution is relatively simple (and we found a number of bold claims online), our proof is surprisingly intricate and mixes combinatorial and geometric arguments; we argue that any rigorous argument must very likely be rather elaborate.Comment: 29 page

Convex minorant trees associated with Brownian paths and the continuum limit of the minimum spanning tree

We give an explicit construction of the scaling limit of the minimum spanning tree of the complete graph. The limit object is described using a recursive construction involving the convex minorants of a Brownian motion with parabolic drift (and countably many i.i.d. uniform random variables); we call it the Brownian parabolic tree. Aside from the new representation, this point of view has multiple consequences. For instance, it permits us to prove that its Hausdorff dimension is almost surely 3. It also intrinsically contains information related to some underlying dynamics: one notable by-product is the construction of a standard metric multiplicative coalescent which couples the scaling limits of random graphs at different points of the critical window in terms of the same simple building blocks. The above results actually fit in a more general framework. They result from the introduction of a new family of continuum random trees associated with functions via their convex minorants, that we call convex minorant trees. We initiate the study of these structures in the case of Brownian-like paths. In passing, we prove that the convex minorant tree of a Brownian excursion is a Brownian continuum ranndom tree, and that it provides a coupling between the Aldous--Pitman fragmentation of the Brownian continuum random tree and its representation by Bertoin.Comment: 56 pages, 2 figure

Presencia de Bryopsida fértil en los niveles Westfalianos del subgrup Itararé, Cuenca de Paraná, Brasil

The bryophyte fossils are rare, mainly in Paleozoic sedimentary rocks in spite of being present since the Silurian Period. In the Division Bryophyta, the fossils that belong to the Class Bryopsida are recognized since the Carboniferous, but they are extremely scarce. They are plentiful only in Permian sediments, in the Petchora, Kuznetsk and Russian Platform basins, also in Antarctica, Karoo basin (the last in South Africa) and India. Identified at the genus Dwykea, gametophyte specimens bearing pleurocarpous sporophyte were recovered from the lowermost levels of Itararé Subgroup, near Campinas city, S. Paulo State. These fossils correspond to the first register of bryophyte female gametophyte for the Carboniferous Period. The microflora in association with these fossils allow correlations of these levels to the Palynozone Ahrensisporites cristatus of Westphalian age. Related to proglacial sediments, they may correspond to a tundra vegetation covering the Northeastern border of Paraná Basin, during the Westphalian.Los fósiles atribuidos a briófitos son escasos, principalmente en rocas paleozoicas a pesar de ser registrados desde el Silúrico. Para la División Bryophyta, los fósiles correspondientes a la Clase Bryopsida comienzan a ser encontrados a partir del Carbonífero, aunque son bastante escasos. Registros más abundantes son conocidos para el Pérmico en rocas de las cuencas de Petchora, Kuznetsk y en la Plataforma Rusa, así como en la Antártica, en la Cuenca del Karoo (África del Sur) e India. Especímenes de gametófitos con esporofitos pleurocárpicos del género Dwykea fueron colectados en los niveles inferiores del Subgrupo Itararé, próximos a la ciudad de Campinas, Estado de S. Paulo. Estos fósiles corresponden al primer registro para el periodo Carbonífero de gametofitos femeninos fértiles. La microflora asociada a los fósiles de Dwykea permite establecer correlaciones con la Palinozona Ahrensisporites cristatus de edad westfaliana. Además los niveles donde fueron colectados los ejemplares de Dwykea corresponden a sedimentos proglaciales, que son interpretados como una vegetación de tundra que habitaba en el margen noreste de la Cuenca del Paraná, durante el Westfaliano

Middle Permian Ostracods from Tak Fa Limestone, Phetchabun Province, Central Thailand

This paper presents the first Permian ostracod fauna discovered in Thailand. The ostracods are recovered from the Tak Fa Limestone (Middle Permian) in Phetchabun province, central Thailand. The ostracods belong to eight genera and 15 species. Four species are newly described: Sargentina phetchabunensis nov. sp., Geffenina bungsamphanensis nov. sp., Reviya subsompongensis nov. sp. and Bairdia takfaensis nov. sp. The ostracod assemblages characterize a shallow marine, near shore environment at the time of deposition. Except for one species, which shows palaeobiogeographical links between Central Thailand and South China, all the other species are endemic

Comparative study of meningitis dynamics across nine African countries: a global perspective

<p>Abstract</p> <p>Background</p> <p>Meningococcal meningitis (MM) represents an important public health problem especially in the "meningitis belt" in Africa. Although seasonality of epidemics is well known with outbreaks usually starting in the dry season, pluri-annual cycles are still less understood and even studied. In this context, we aimed at study MM cases time series across 9 sahelo-sudanian countries to detect pluri-annual periodicity and determine or not synchrony between dynamics. This global and comparative approach allows a better understanding of MM evolution in time and space in the long-term.</p> <p>Results</p> <p>We used the most adapted mathematical tool to time series analyses, the wavelet method. We showed that, despite a strong consensus on the existence of a global pluri-annual cycle of MM epidemics, it is not the case. Indeed, even if a clear cycle is detected in all countries, these cycles are not as permanent and regular as generally admitted since many years. Moreover, no global synchrony was detected although many countries seemed correlated.</p> <p>Conclusion</p> <p>These results of the first large-scale study of MM dynamics highlight the strong interest and the necessity of a global survey of MM in order to be able to predict and prevent large epidemics by adapted vaccination strategy. International cooperation in Public Health and cross-disciplines studies are highly recommended to hope controlling this infectious disease.</p

EVIDENCE OF A GUADALUPIAN AGE FOR THE KHUFF FORMATION OF SOUTHEASTERN OMAN: PRELIMINARY REPORT

The Guadalupian succession of the Huqf area (Sultanate of Oman) represents a mega-sequence comprising the fluvial terrigenous Gharif Formation and the overlying marine Khuff Formation. The Khuff Fm. is subdivided into four members and is composed of marls and bioclastic limestones. The Khuff Fm. yields a rich fauna of brachiopods, conodonts, foraminifers, bivalves, gastropods, ostracods and cephalopods. The brachiopod fauna of the Khuff Fm. includes strophomenids, productids, orthids and terebratulids. The associated conodont fauna includes Hindeodus excavatus Behnken, Merrilina sp., M. praedivergens Kozur &amp; Mostler, and Sweetina n. sp. (systematic descriptions of conodonts are given in the Paleontological Appendix). Foraminifers are represented by species of Miliolina and Rotaliina. The Khuff Fm. is given a Wordian age, based on brachiopods and conodonts. The depositional environment of the Khuff Fm. of southeastern Oman corresponds to the outer shelf of a large carbonate platform covering most of the Arabian Platform. The Khuff Fm. is interpreted as a major transgressive-regressive cycle related to differential subsidence.&nbsp